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Ee = ∫ Ω ψe (ε, s) dV . (8) The constitutive behaviour is determined by the definition of the strain energy density ψe (ε, s) = ψe− (ε) + g(s)ψe+ (ε) . (9) The strain energy density is decomposed in a crack driving, strain part ψe+ that is affected by the degradation function g(s) and a part that is associated with compressive strain states ψe−.
The compressive strain energy is not affected by g(s) which models the impenetrability of cracks during crack closure, i.e. no degradation of the 'compressive' stress see (17).
The stress is σ = ∂ψe ∂ε = ∂ψe− ∂ε + g(s)∂ψe+ ∂ε . (17) The wave speed of dilatational waves in the considered elastic medium is cd = √λ + 2µ ρ . (18) The total kinetic energy of the body is assumed not to be affected by the phase field, i.e K( ˙u) = ∫ Ω 1 2ρ ˙u · ˙u dV . (19) The work of external forces acting on the boundary ∂Ω reads P = ∫ ∂Ωt t∗ · u dA. (20) Finally, the dynamic fracture problem can be stated using Hamilton's principle δ t2∫ t1 L dt = 0, (21) for arbitrary times t1 < t2, where the Lagrangian is given by L = K − (Ee + Es − P). (22) For the problem at hand, the Euler-Lagrange equations following from (21) are the equation of motion ρ¨u − divσ = 0, (23) Applied Mechanics and Materials Vol. 869 103and the phase field equation g'(s)ψ+e − Gc [ 2ϵ∆s + 1 − s 2ϵ ] = 0 (24) as well as the Neumann boundary conditions for the displacement σn = t∗ on ∂Ωt (25) and for the phase field ∇s · n = 0 on ∂Ω. (26) In (24) it becomes apparent, that the property
The factor δ further reduces the allowable time step and accounts for nonlinear effects.
Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis" References [1] Ambati, M., Gerasimov, T., Lorenzis, L. (2015).

The efficient photocatalytic H2 production has been majorly relies on the properties of the photocatalyst used for the reaction.
Nevertheless it is apparently reveals that H2 production efficiency is relying on the properties of semiconductor TiO2 photocatalyst.
Hawai, Catalytic properties of ruthenium oxide on n-type semiconductors under illumination, J.
Ma, Photoelectrical and charge transfer properties of hydrogen evolving TiO2 nanotube arrays electrodes annealed in different gases, Int.
Lee, Preparation of highly ordered cubic mesoporous WO3/TiO2 films and their photocatalytic properties, Chem.

The reactive diffusion process is followed to grow tungsten disilicides for the use as integrated circuits due to their beneficial properties of low electrical resistivity and good thermal stability.
Other than the importance of understanding the diffusion-controlled growth process, the estimation of the diffusion coefficients is important to understanding many physical and mechanical properties of materials.
Bulk diffusion couple experiments are important for an extensive analysis of the diffusion process without the influence of other factors.
On the other hand, the growth kinetics is affected when it grows along with other phases.
Study on diffusion is important for developing indepth understanding of many properties.